Lecture Outline Crystallography

Transcription

1 Lecture Outline Crystallography Short and long range Order Poly- and single crystals, anisotropy, polymorphy Allotropic and Polymorphic Transitions Lattice, Unit Cells, Basis, Packing, Density, and Crystal Structures Points, Directions, and Planes in the Unit Cell Structures with more than one atoms per lattice point Interstitial Sites Ionic Crystal Structures Covalent Crystal Structures Metallic Crystal structures Diffraction and modern microscopy 1

2 Lowest ENERGY state at each temperature Non dense, random packing There is still short range order typical neighbor bond energy Energy typical neighbor bond length r Dense, regular packing Energy There is long range order in addition to short range order typical neighbor bond energy typical neighbor bond length r Dense, regular-packed structures tend to have lower energy, of course everything depends on temperature, at low enough temperatures many materials crystallize

3 MATERIALS AND PACKING Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics - semiconductors -some polymers Glass is actually a cold melt, over long enough times, it crystallizes, old church windows are thicker at the bottom Noncrystalline materials... atoms have no periodic packing occurs for: -complex structures - rapid cooling Si One of many forms of crystalline SiO Oxygen "Amorphous" = Noncrystalline noncrystalline SiO Quartz glass, a cold melt 3

4 Classification of materials based on type of atomic order. 4

5 Levels of atomic arrangements in materials: (a)inert monoatomic gases have no regular ordering of atoms: (b,c) Some materials, including water vapor, nitrogen gas, amorphous silicon and silicate glass have shortrange order. (d) Metals, alloys, many ceramics, semiconductors and some polymers have regular ordering of atoms/ions that extends through the material = crystals 5

6 Basic Si-0 tetrahedron in silicate glass. X-ray diffraction shows only short range order. Note that this cannot exist in quartz at room temperature, there the tetrahedron is distorted resulting in pronounced anisotropy effects such as piezoelectricity. In ß-cristobalite, a high temperature phase of quartz, we have an undistorted tetrahedron again, X-ray diffraction shows long range order 6

7 Again there are different crystallograp hic phases, i.e. long range order structures until the crystal melts and only short range order remains 7

8 Atomic arrangements in (a) Amorphous silicon with H (b) Crystalline silicon. Note the variation in the interatomic distance for amorphous silicon, but there still is short-range order 8

9 POLY-CRYSTALS Most engineering materials are poly-crystalline! 1 mm Nb-Hf-W plate with an electron beam weld. Each "grain" is a single crystal. If crystals are randomly oriented, component properties are not directional, but frequently we have texture, preferred orientation of poly-crystals resulting in pronounced anisotropy Crystal sizes range from 1 nm to cm, (i.e., from a few to millions of atomic layers). 9

10 SINGLE versus POLY-CRYSTALS Single (Mono-)crystals - Properties vary with direction: anisotropy. - Example: the modulus of elasticity (E) in BCC iron: E (diagonal) = 73 GPa Poly-crystals - Properties may/may not vary with direction, depending on degree of texture. - If grains are randomly oriented: isotropic. (E poly iron = 10 GPa) - If grains are textured, anisotropic. E (edge) = 15 GPa 00 µm 10

11 Allotropic and Polymorphic Transitions Allotropy - The characteristic of an element being able to exist in more than one crystal structure, depending on temperature and pressure. Polymorphism - Compounds exhibiting more than one type of crystal structure. Everything depends on temperature and pressure, e.g. coefficient of thermal expansion can, therefore, only be defined over a certain region of temperature 11

12 Covalently bonded layer Cubic crystal Covalently bonded network of atoms Layers bonded by van der Waals bonding Covalently bonded layer Hexagonal unit cell (a) Diamond unit cell (b) Graphite The FCC unit cell of the Buckminsterfullerene crystal. Each lattice point has a C 60 molecule Buckminsterfullerene (C 60 ) molecule (the "buckyball" molecule) (c) Buckminsterfullerene 1

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14 U(X) = PE = mgh Metastable Unstable (Activated State) E A U U A* U A U B A A* B Stable X X A X A* X B System Coordinate, X = Position of Center of Mass Fig. 1.7: Tilting a filing cabinet from state A to its edge in state A* requires an energy E A. After reaching A*, the cabinet spontaneously drops to the stable position B. PE of state B is lower than A and therefore state B is more stable than A. 14

15 Example: heating and cooling of a hanging iron wire Demonstrates "polymorphism" Temperature, C The same group of atoms has more than one crystal structure. The actual structure depends on temperature and pressure Liquid BCC Stable longer 914 Tc 768 FCC Stable BCC Stable heat up cool down shorter! longer! magnet falls off shorter 15

16 Lattice, Unit Cells, Basis, and Crystal Structures Lattice - a 3D collection of points that divide space into smaller equally sized units. Basis - a group of atoms associated with a lattice point. This may be one single atom or a group of atoms. Unit cell - a subdivision of the lattice that still retains the overall characteristics of the entire lattice, contains at least one atom may contain many atoms. Atomic radius - apparent radius of an atom, typically calculated from the dimensions of the unit cell, using close-packed directions (depends upon type of bonding, coordination number, quantum mechanics). Packing factor - The fraction of space in a unit cell occupied by atoms. 16

17 The fourteen types of Bravais) lattices grouped in seven crystal systems: triclinic monocline rhombohedral = (trigonal) orthorhombic tetragonal hexagonal cubic 17

18 Definition of lattice parameters in cubic, orthorhombic, and hexagonal crystal systems. For cubic crystals, however, calculations are just like with Cartesian coordinates Note that angles are not always 90 degrees and coordination axis lengths are not necessarily all equal, as you know them to be from Cartesian coordinates 18

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20 Lattice Points and Directions in the Unit Cell Miller-indices - A shorthand notation to describe certain crystallographic directions and planes in a material. Lattice directions are in direct space and are denoted by [ ] brackets. A negative number is represented by a bar over the number. Directions of a form (also called family) - Crystallographic directions that all have the same characteristics, although their sense may be different. Denoted by <> brackets, they are symmetrically equivalent 0

21 Lattice Planes in the Unit Cell are an altogether different matter! Miller-indices - A shorthand notation to describe certain crystallographic directions and planes in a material. Lattice planes are represented by the vector that is normal (perpendicular to them), these are 3D vectors in reciprocal (or dual) space (reciprocal space is nothing fancy - it is just a mathematical convenience!) Directions of a form (also called family) lattice planes that all have the same characteristics, although their sense may be different. Denoted by {} brackets, they are symmetrically equivalent. Now if the lattice point represents more than one point the front side and the back side of one and the same plane may have very different chemical properties as different atoms will be exposed, e.g. ZnS structure 1

22 We start with the coordinates of lattice points in order to define the Miller indices of lattice directions Coordinates of selected points in the unit cell. The number refers to the distance from the origin in terms of lattice parameters.

23 Determining Miller Indices of Directions Determine coordinates of two points that lie in direction of interest, u 1 v 1 w 1 and u v w calculations are simplified if the second point corresponds with the origin of the coordinate system Subtract coordinates of second point from those of first point u = u 1 -u, v = v 1 -v, w = w 1 -w Clear fractions from the differences to give indices in lowest integer values. Write indices in [] brackets. Negative integer values are indicated with a bar over the integer, [uvw] and [uvw] are running in opposite directions 3

24 Direction A 1. Two points are 1, 0, 0, and 0, 0, 0. 1, 0, 0, - (0, 0, 0) = 1, 0, 0 3. No fractions to clear or integers to reduce 4. [100] Direction B 1. Two points are 1, 1, 1 and 0, 0, 0. 1, 1, 1, - (0, 0, 0) = 1, 1, 1 3. No fractions to clear or integers to reduce 4. [111] Direction C 1. Two points are 0, 0, 1 and 1/, 1, 0. 0, 0, 1 (1/, 1, 0) = -1/, -1, 1 3. (-1/, -1, 1) = -1,-, 4. [ 1 ] 4

25 Equivalency of crystallographic directions of a form in cubic systems. 5

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27 Determining Miller Indices of Planes Identify the coordinate intersects of the plane, if plane is parallel to one of the axes, this intercept is taken to be infinite Take the reciprocal of the intercept Clear fractions, but do not reduce to lowest integers Cite in (h k l) parentheses Negative integer values are indicated with a bar over the integer (h k l) is the same plane as (h k l), just its back side 7

28 Plane A 1. x = 1, y = 1, z = 1.1/x = 1, 1/y = 1,1 /z = 1 3. No fractions to clear 4. (111) Plane B 1. The plane never intercepts the z axis, so x = 1, y =, and z =.1/x = 1, 1/y =1/, 1/z = 0 3. Clear fractions: 1/x =, 1/y = 1, 1/z = 0 4. (10) Plane C 1. We shall move the origin, since the plane passes through 0, 0, 0. Let s move the origin one lattice parameter in the y-direction. Then, x =, y = -1, and z =.1/x = 0, -1/y = -1, 1/z = 0 3. No fractions to clear. 4.(010) that seemed a bit arbitrary, we could have moved the origin in the y direction as well, then we would have gotten (010), which is just the back side of (010) 8

29 x intercept at a / z intercept at x z Unit cell b a c y Miller Indices (hkl ) : y intercept at b (a) Identification of a plane in a crystal z (010) (010) z (010) 1 (10) (010) x (010) y x (001) (110) y (100) z z (110) (111) (111) y y x y x (b) Various planes in the cubic lattice z 9

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31 Drawing Direction and Plane [11] [10] Draw (a) the direction and (b) the plane in a cubic unit cell. Construction of a (a) direction and (b) plane within a unit cell 31

32 SOLUTION a. Because we know that we will need to move in the negative y-direction, let s locate the origin at 0, +1, 0. The tail of the direction will be located at this new origin. A second point on the direction can be determined by moving +1 in the x-direction, in the negative y-direction, and +1 in the z direction. [10] b. To draw in the plane, first take reciprocals of the indices to obtain the intercepts, that is: x = 1/- = -1/ y = 1/1 = 1 z = 1/0 = Since the x-intercept is in a negative direction, and we wish to draw the plane within the unit cell, let s move the origin +1 in the x-direction to 1, 0, 0. Then we can locate the x-intercept at 1/ and the y-intercept at +1. The plane will be parallel to the z-axis. 3

33 Determining Miller-Bravais Indices for Planes and Directions in hexagonal system Miller-Bravais indices are obtained for crystallographic planes, directions, and points in hexagonal unit cells by using a four-axis coordinate system. For planes (hkil), the index i = -(h+k), i.e. h+k = -i For directions [uvtw], we have also t = (u+v), i.e. u+v = -t 33

34 Miller-Bravais indices for planes are straightforward, just as we obtained the intersects for 3 axes, we have to obtain them now for 4 axes SOLUTION Plane A 1. a 1 = a = a 3 =, c = 1. 1/a 1 = 1/a = 1/a 3 = 0, 1/c = 1 3. No fractions to clear 4. (0001) Plane B 1. a 1 = 1, a = 1, a 3 = -1/, c = 1. 1/a 1 = 1, 1/a = 1, 1/a 3 = -, 1/c = 1 3. No fractions to clear 4. (111) 34

35 Determining directions in the hexagonal system is a bit more challenging it is easier to calculate with 3 indices and then simply make up the forth SOLUTION (Continued) Direction C 1. Two points are 0, 0, 1 and 1, 0, 0.. 0, 0, 1, - (1, 0, 0) = -1, 0, 1 3. No fractions to clear or integers to reduce. 4. [101] or [113] Direction D 1. Two points are 0, 1, 0 and 1, 0, 0.. 0, 1, 0, - (1, 0, 0) = -1, 1, 0 3. No fractions to clear or integers to reduce. 4. extension to 4 indices looks easy, but is not! [110]or[1100] How did we get the forth index? All have to be relabeled, say [UVW] are the three indexes, u = 1 / 3 (U V), v = 1 / 3 (V-U), t = - 1 / 3 (u+v), w = W 35

36 Miller-Bravais Indices of important directions Typical directions in the hexagonal unit cell, using both threeand-four-axis systems. The dashed lines show that the [110] direction is equivalent to a [010] direction. Densely packed lattice directions in the basal plane (0001), e.g. [100], [010], and [110] have similar Miller-Bravais indices, important for dislocation slip systems 36

37 37 Now as we have coordinates for lattice points, lattice directions and lattice planes, we can start making crystallographic calculations Good news: everything is easy in the cubic system angle between two different direct [uvw] (or reciprocal (hkl)) space directions So angle between planes is calculated as angle between their normals, if you were to use a protractor (or contact goniometer), you would just measure an angle ß arccos z v u z v u z z v v u u = α arccos l k h l k h l l k k h h = β

38 Bad news: everything gets a bit more difficult in non cubic systems for two reasons, the coordinate axes are no longer perpendicular to each other, the unit vectors of coordinate axes have different length No problem: we have the metric tensor G, a 3 by 3 matrix a abcos γ accos β abcos γ b bccosα accos β bccosα c If cubic simply a a a So angle between lattice directions become α = arccos u' Gv u' Gu v' Gv 38

39 39 Example: what is the angle between [100] and [111] in (tetragonal) ß-Sn? a = nm, c =0.318nm, Solution: 0) 0 ( ) 0 1 ( a c a a c a a = = nm a Gu u 583 ' = 0. = nm c a a G ) 1 (1 = + + =

40 Putting into relation from above α = arccos u' Gv u' Gu v' Gv α = a arccos = 7. a So that was not all that difficult!!! 40

41 Same basic idea for lattice planes, i.e. their normals which are vectors in reciprocal (or dual) space everything is easy in cubic systems, in other crystal systems we use reciprocal metric tensor, the reciprocal 3 by 3 matrix of the metric tensor b c sin α abc (cosα cos β cos γ ) ab c(cosα cos γ cos β abc a (cosα cos β cos γ a c sin β bc(cos β cos γ cosα ab a c(cosα cos γ bc(cos β cos γ a b sin γ cos β cosα If cubic simply, this also defines the reciproc al lattice 1 a a a α So angle between normals of lattice planes become = arccos h' Gh h' Gk k' Gk 41

42 4 Metric tensor (and its reciprocal) also define reciprocal lattice a b a b c c a c a b c b c b a = = = = = = ) sin ( * ) sin ( * ) sin ( * γ β α It is just a mathematical convenience, particularly useful for interpretation of (X-ray and electron diffraction) data d* = H = h a 1 * + ka * + l a 3 *= 1 / d In cubic systems simply: l k h a d hkl + + =

43 Other geometric considerations, valid for all crystal systems, that is the great thing about Miller indices!!! When does a direction lie in a plane? If their dot product is zero, e.g. and (111), = 0 [ 1 01] (1 31) [1 01] But does not lie in as ? 0 What are the indices of the direction that mediated the intersection of two plane? We have to determine cross products ( 1 3) [53] (111) x = 43

44 Centered lattices and structures with more than one lattice point per unit cell (a) Illustration showing sharing of face and corner atoms. (b) The models for simple cubic (SC), body centered cubic (BCC), and facecentered cubic (FCC) unit cells, assuming only one atom per lattice point. 44

45 1 /8 th of an atom R Half of an atom a R R a a Fig. 1.38: The FCC unit cell. The atomic radius is R and the lattice parameter is a 45

46 Determining the Number of Lattice Points in Cubic Crystal Systems Determine the number of lattice points per cell in the cubic crystal systems. If there is only one atom located at each lattice point, calculate the number of atoms per unit cell. SOLUTION In the SC unit cell: lattice point / unit cell = (8 corners)1/8 = 1 In BCC unit cells: lattice point / unit cell = (8 corners)1/8 + (1 center)(1) = In FCC unit cells: lattice point / unit cell = (8 corners)1/8 + (6 faces)(1/) = 4 If there is only on atom per lattice point, as in typical metal structures, the number of atoms per unit cell would be 1,, and 4, for the simple cubic, body-centered cubic, and facecentered cubic, unit cells, respectively. But there are also many structures with more than one atom per lattice point 46

47 Determining the Relationship between Atomic Radius and Lattice Parameters Determine the relationship between the atomic radius and the lattice parameter in SC, BCC, and FCC structures when one atom is located at each lattice point. The relationships between the atomic radius and the lattice parameter in cubic systems 47

48 SOLUTION Referring to Figure above, we find that atoms touch along the edge of the cube in SC structures. a0 = r In BCC structures, atoms touch along the body diagonal. There are two atomic radii from the center atom and one atomic radius from each of the corner atoms on the body diagonal, so a 0 = 4r In FCC structures, atoms touch along the face diagonal of the cube. There are four atomic radii along this length two radii from the face-centered atom and one radius from each corner, so: a 0 = 3 4r 48

49 THEORETICAL DENSITY, ρ # atoms/unit cell Atomic weight (g/mol) Volume/unit cell (cm 3 /unit cell) ρ = n A V c N A Avogadro's number (6.03 x 10 3 atoms/mol) Example: Copper crystal structure = FCC: 4 atoms/unit cell atomic weight = g/mol (1 amu = 1 g/mol) atomic radius R = 0.18 nm (1 nm = 10-7 cm) Vc = a 3 ; For FCC, a = 4R/ ; Vc = 4.75 x 10-3 cm 3 Result: theoretical ρcu = 8.89 g/cm 3 Compare to actual: ρcu = 8.94 g/cm 3 So there must be some real structure in addition! 49

50 Determining the Density of BCC Iron Determine the density of BCC iron, which has a lattice parameter of nm. SOLUTION Atoms/cell =, a 0 = nm = cm Atomic mass = g/mol 3 a 0 Volume of unit cell = = ( cm) 3 = cm 3 /cell Avogadro s number N A = atoms/mol Density ρ = ρ = (3.54 (number of (volume ()(55.847) 10 of = 7.88g / 4 3 )(6.0 atoms/cell)(atomic unit cell)(avog adro' 10 ) mass s of number) cm 3 iron) 50

51 DENSITIES OF MATERIAL CLASSES Metals typically have close-packing (metallic bonding) large atomic mass Ceramics have less dense packing (covalent bonding) often lighter elements Polymers have... poor packing (parts often amorphous) lighter elements (C,H,O) Composites have... intermediate values ρ (g/cm 3 ) Metals/ Alloys Platinum Gold, W Tantalum Silver, Mo Cu,Ni Steels Tin, Zinc Titanium Aluminum Magnesium Graphite/ Ceramics/ Semicond Polymers Composites/ fibers Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers in an epoxy matrix). Zirconia Al oxide Diamond Si nitride Glass-soda Concrete Silicon Graphite PTFE Silicone PVC PET PC HDPE, PS PP, LDPE Glass fibers GFRE* Carbon fibers CFRE* Aramid fibers AFRE* Wood 51

52 ATOMIC PACKING FACTOR APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres APF for a simple cubic structure = 0.5 a close-packed directions contains 8 x 1/8 = 1 atom/unit cell R=0.5a atoms unit cell APF = 1 volume 4 atom 3 π (0.5a)3 a 3 volume unit cell 5

53 ATOMIC PACKING FACTOR: BCC APF for a body-centered cubic structure = 0.68 R a atoms unit cell APF = Close-packed directions: length = 4R = 3 a Unit cell contains: x 1/8 = atoms/unit cell 4 3 π ( 3a/4)3 a 3 volume unit cell volume atom 53

54 ATOMIC PACKING FACTOR: FCC APF for a body-centered cubic structure = 0.74 a atoms unit cell APF = Close-packed directions: length = 4R = a Unit cell contains: 6 x 1/ + 8 x 1/8 = 4 atoms/unit cell π ( a/4)3 a 3 volume unit cell That is the highest possible packing factor!! volume atom 54

55 The hexagonal close-packed (HCP) structure (left) and its unit cell. The packing factor is also 74 %, i.e. the highest possible, just as we had for FCC spheres 55

56 Stacking sequences of closed packed structures, i.e. those with 74 % Packing factor The ABABAB stacking sequence of close-packed planes produces the HCP structure. 56

57 The ABCABCABC stacking sequence of closepacked planes produces the FCC structure. 57

58 Coordination and nearest neighbors Illustration of coordinations in (a) SC- six fold and (b) BCC 8 fold unit cells. Six atoms touch each atom in SC, while the eight atoms touch each atom in the BCC unit cell. 58

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60 Lattice Crystal Basis a 90 a Unit cell (a) (b) (c) Unit cell (d) Basis placement in unit cell (0,0) y (1/,1/) x Fig. 1.70: (a) A simple square lattice. The unit cell is a square with a side a. (b) Basis has two atoms. (c) Crystal = Lattice + Basis. The unit cell is a simple square with two atoms. (d) Placement of basis atoms in the crystal unit cell. 60

61 M.C. Escher s two dimensional patterns are filling the whole D space. We can determine a unit cell which happened to be composed off many details, regardless of which periodic detail we use for the choice of the unit cell, these cells will always be identical. So we can think about it in a way that all the details are shrunk into a lattice points and all that remains is the geometrical array of lattice points we had in the previous slide 61

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67 Interstitial Sites Interstitial sites - Locations between the normal atoms or ions in a crystal into which another - usually different - atom or ion is placed. Typically, the size of this interstitial location is smaller than the atom or ion that is to be introduced. Cubic site - An interstitial position that has a coordination number of eight. An atom or ion in the cubic site touches eight other atoms or ions. Octahedral site - An interstitial position that has a coordination number of six. An atom or ion in the octahedral site touches six other atoms or ions. Tetrahedral site - An interstitial position that has a coordination number of four. An atom or ion in the tetrahedral site touches four other atoms or ions. 67

68 The location of the interstitial sites in cubic unit cells. Only representative sites are shown. 68

69 Interstitial sites are important because we can derive more structures from these basic FCC, BCC, HCP structures by partially or completely different sets of these sites 69

70 Calculating Octahedral Sites How many octahedral sites are there in one FCC unit cell? SOLUTION The octahedral sites include the 1 edges of the unit cell, with the coordinates 1 0, 1 0,0,,0,0,0 1 1,1,0 1 1,,0 1 1,0, 1,0,1 1 1,,1 1 1,1, plus the center position, 1/, 1/, 1/. 1,1,1 1 0,,1 1 0,1, 70

71 SOLUTION (Continued) Each of the sites on the edge of the unit cell is shared between four unit cells, so only 1 / 4 of each site belongs uniquely to each unit cell. Therefore, the number of sites belonging uniquely to each cell is: (1 edges) ( 1 / 4 per cell) + 1 center location = 4 octahedral sites 71

72 How much space is there available to fill the different sites with hard sphere atoms? 7

73 The radios ratios have to be of course the same as the once we discussion in connection with coordination 73

74 Crystal Structures of Ionic Materials Factors that need to be considered in order to understand crystal structures of ionically bonded solids: Ionic Radii for filling of various interstitial sites Electrical Neutrality as a structural principle 74

75 STRUCTURE OF NaCl Compounds: Often have close-packed structures there is more than one atom per lattice point. Structure of NaCl FCC + octahedral Interstitial site filled 75

76 Na + Cl Fig. 1.36: A possible reduced sphere unit cell for the NaCl (rock salt) crystal. An alternative unit cell may have Na+ and Cl interchanged. Examples: AgCl, CaO, CsF, LiF, LiCl, NaF, NaCl, KF, KCl, MgO 76

77 Simple cubic packing with the cube interstitial site filled by another atom, note that there are two atoms per lattice point, i.e. the lattice as such is still simple cubic Cl Cs + Fig. 1.37: A possible reduced sphere unit cell for the CsCl crystal. An alternative unit cell may have Cs + and Cl interchanged. Examples: CsCl, CsBr, CsI, TlCl, TlBr, TlI. 77

78 (a) The zinc blende unit cell, (b) plan view. There is usually a large covalent contribution to these bonds. The coordination is quite low for ionic bonds. 78

79 (a) Fluorite unit cell, (b) plan view. All 8 tetrahedral interstitial sites are filled by F anions, as the formula is CsF, there is three atoms per lattice point. U), ThO and ZrO have the same structure, there is also the antifluorite structure for Li O, Na O, K O with cations and anions reversed 79

80 g Corundum structure of alpha-alumina (a-ai 0 3 ), O in HCP and Al in some of the octahedral interstitial sites, possibly the most widely used ceramic 80

81 Covalent Structures Covalently bonded materials frequently have complex structures in order to satisfy the directional restraints imposed by the bonding. Diamond cubic (DC) - A special type of facecentered cubic crystal structure found in carbon, silicon, α-sn, and other covalently bonded materials. 81

82 C a a a Fig. 1.33: The diamond unit cell is cubic. The cell has eight atoms. Grey Sn (α-sn) and the elemental semiconductors Ge and Si have this crystal structure. 8

83 S a Zn a a Fig. 1.34: The Zinc blende (ZnS) cubic crystal structure. Many important compound crystals have the zinc blende structure. Examples: AlAs, GaAs, GaP, GaSb, InAs, InP, InSb, ZnS, ZnTe. 83

84 The silicon-oxygen tetrahedron and the resultant ß- cristobalite form of silica. Note: there is six atom at every lattice point, i.e. one Si in the center, 4 O within the tetrahedron and 4 times ¼ Si at the apexes of each tetrahedron 84

85 The unit cell of crystalline polyethylene, Orthorhombic, there are typically many defects in polymer crystals 85

86 METALLIC CRYSTALS tend to be densely packed. have several reasons for dense packing: -Typically, only one element is present, so all atomic radii are the same. -Metallic bonding is not directional. -Nearest neighbor distances tend to be small in order to lower bond energy. have the simplest crystal structures. We will revisit the two closest packed structures and give some examples for each of them 86

87 (a) FCC Unit Cell R a a a (b) a (c) Fig. 1.30: (a) The crystal structure of copper is Face Centered Cubic (FCC). The atoms are positioned at well defined sites arranged periodically and there is a long range order in the crystal. (b) An FCC unit cell with closed packed spheres. (c) Reduced sphere representation of the FCC unit cell. Examples: Ag, Al, Au, Ca, Cu, γ-fe (>91 C), Ni, Pd, Pt, Rh 87

88 Layer B (a) a (c) Layer A Layer B Layer A c Layer A (d) (b) Examples: Be, Mg, α-ti ( < 88 C ), Cr, Co, Zn, Zr, Cd Layer A Be very careful, the yellow and blue spheres represent one and the same type of atoms Fig. 1.3: The Hexagonal Close Packed (HCP) Crystal Structure. (a) The Hexagonal Close Packed (HCP) Structure. A collection of many Zn atoms. Color difference distinguishes layers (stacks). (b) The stacking sequence of closely packed layers is ABAB (c) A unit cell with reduced spheres (d) The smallest unit cell with reduced spheres. 88

89 What is steel? It s not an element, it s an intestinal alloy, about 1 C atom every 50 Fe atoms, i.e. 0.4 weight % C So the crystal structure is BCC, as it is Fe at room temperature, every now and then a much smaller c atom fits into one of the cubic interstitial sites and this increases the strength from about 15 MPa to more than 1500 MPa How is that little amount of C accomplishing so much? More about things such as this in next lecture on lattice defects 89

90 SUMMARY Crystals Atoms typically assemble into crystals some materials e.g. glass are amorphous, i.e. have only short range order We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but properties are frequently quite non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains as e.g. in steel and other engineering materials 90

91 Diffraction Techniques for Crystal Structure Analysis Diffraction - The constructive interference, or reinforcement, of a beam of x-rays or electrons interacting with a material. The diffracted beam provides useful information concerning the structure of the material. Bragg s law - The relationship describing the angle? at which a beam of x-rays of a particular wavelength diffracts from crystallographic planes of a given interplanar spacing. In a diffractometer a moving x-ray detector records the? angles at which the beam is diffracted, giving a characteristic diffraction pattern 91

92 Lattice constants range from about 0.1 nm to some nm. Most crystals lattice constants are in the 1-5 nm range, since crystals are periodic, they are the ideal diffraction grating for X-rays, Max von Laue s 1914 Nobel prize that proved both, crystals are periodic 3D arrangements of atoms and X-rays are waves (well wave-particles actually) 9

93 The diffraction angle is always? Bragg s law: n l = d hkl sin? n the order of reflection can be dropped by allowing indices for hkl that are not smallest integers, i.e. HKL 93

94 Single crystal diffractometers are more sophisticated 94

95 Powder diffractometers (for polycrystalline powder) are less sophisticated and can be found in many places 95

96 (a) sketch of a diffractometer viewed from above, showing powder sample, incident and diffracted beams. (b) diffraction pattern obtained from a sample of gold powder, notice the Miller indices of the diffracting crystal planes 96

97 1 detector incoming X-rays extra distance travelled by wave θ 1 θ outgoing X-rays λ d reflections must be in phase to detect signal spacing between planes Measurement of: reflection angles (Bragg angles), θ, for X-rays provide atomic spacing, d. x-ray intensity (from detector) θc d=nλ/sinθc θ 97

98 (a) Destructive and (b) reinforcing (constructive) interference between x-rays reflected on a set of planes. Reinforcement occurs only at angles that satisfy Bragg s law. l = d HKL sin? No other X-ray diffraction peaks occur 98

99 X-ray Diffraction for an forensic examination The results of a x-ray diffraction experiment on some metal powder found at a crime scene using characteristic x-rays with? = Å (a radiation obtained from molybdenum (Mo) target) show that diffracted peaks occur at the following? angles: Determine the indices of the plane producing each peak, and from that the lattice parameter of the material and from that identify the material (you know it is some metal so it is a good guess to assume the crystals in the powder are cubic) 99

100 SOLUTION λ = d HKL sin? d HKL = H 1 + K + L λ ( ) = d HKL sin θ = ( Ha * + Kb* + Lc*) = ( d ) 1 * 1 HKL = r g We first determine the sin? value for each peak, then divide through by the lowest denominator, , guess the indices 100

101 SOLUTION (Continued) We could then use? values for any of the peaks to calculate the interplanar spacing and thus the lattice parameter. Picking peak 8:? = 59.4 or? = 9.71 d 400 = λ sin θ = sin(9.71) = Å a 0 = d 400 h + k + l = ( )(4) =.868 Å This is the lattice parameter for body-centered cubic iron. So the gardener did steal the cookies only kidding Since out of all metals only Po has the simple cubic structure, and we do know our material was not radioactive and difficult to obtain, we didn t check for an indexing scheme for a single cubic lattice, it would have gotten us nowhere anyway 101

102 There are certain systematic absences of reflections You may have noticed that there are no (hkl) triplets with h+k+l = odd. Laue s kinematical theory of X-ray diffraction (191) explains why, different structures have forbidden reflections, i.e. reflections that do not show up in diffraction but these planes do of course exist in the crystals, it is just that diffraction on them is destructive Simple cubic crystals have no forbidden reflections 10

103 A TEM micrograph of an aluminum alloy (Al-7055) sample. The diffraction pattern at the right shows large bright spots that represent diffraction from the main aluminum matrix grains. The smaller spots originate from the nano-scale crystals of another compound that is present in the aluminum alloy. 103

104 (In,Ga)P, an important semiconductor for visible light lasers can show a variety of atomic ordering The very same thing, i.e. atomic ordering exist in metals as well 104

105 II-VI semiconductors with atomic ordering, (Cd,Zn)Se quantum dots in ZnSe, [001] plan view (Cd,Mn,Zn)Se quantum dots in <110> cross section 105

106 Unknown crystallographic phase in (In,Ga)(Sb,As) quantum dots in InAs matrix, a Fourier transform power spectrum of a high resolution TEM images replaces frequently a diffraction pattern as much of the information is contained in such calculated images 106

107 False color convergent beam electron diffraction pattern, From the symmetry of the lines (Kikuchi lines), the crystal system can be determined From the fine structure of the diffraction disk, the space group can be determined Works for extremely tiny specs of matter, perhaps nm across 107

108 Wow, what is that? It can t be a crystal as it obviously has a five fold symmetry, it is not amorphous either as it clearly has a diffraction pattern? It s a quasi-crystal, i.e. and entity with short range order (as amorphous and crystalline materials) and long range order (as a crystal), the crucial difference is, the long range order is non periodic 108

109 Crystallographic work can also be done in modern SEMs! Electron backscatter (Kikuchi) diffraction (EBSD), here of a Si crystal 109

110 scanning probe microscopy, STM, AFM, Atoms can be arranged and imaged! Carbon monoxide molecules arranged on a platinum (111) surface. Iron atoms arranged on a copper (111) surface. These Kanji characters represent the word atom. Something more useful, the square of the quantum mechanical wave function of an electron that is trapped in a cage of Cu atoms 110